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Stream: learning: questions

Topic: differentiation/integration

Scott Lingran (Sep 17 2020 at 13:32):

What's the category theoretic understanding of differentiation and integration in calculus?

Right now I understand them as 2-morphisms in the category of Set, which transforms functions into other functions and is isomorphic. Would love any thoughts or if you're able to point me to any materials.

John Baez (Sep 17 2020 at 13:57):

There are no 2-morphisms in the category $\mathsf{Set}$, since $\mathsf{Set}$ is a not a 2-category.

I think you're saying something like this: if $D(\mathbb{R})$ is the set of differentiable functions $f: \mathbb{R} \to \mathbb{R}$ and $\mathbb{R}^\mathbb{R}$ is the set of all functions $f : \mathbb{R} \to \mathbb{R}$, then there is a function

$\displaystyle{ \frac{d}{dx} : D(\mathbb{R}) \to \mathbb{R}^\mathbb{R} }$

sending any function to its derivative.

John Baez (Sep 17 2020 at 13:58):

There's a similar statement for integration. There are many ways to make these statements more complicated and powerful, which are studied in the subject called real analysis.

John Baez (Sep 17 2020 at 13:58):

There are also other ways to think about derivatives and integrals using more category theory.

Nathanael Arkor (Sep 17 2020 at 14:04):

You might be interested in the work on Cartesian differential categories. The category of smooth maps $\mathbb R^n \to \mathbb R^m$ is a motivating example.

James Wood (Sep 17 2020 at 14:58):

Did these authors ever cover integration? I feel as if I have a decent abstract understanding of differentiation, but that understanding is perhaps in conflict with integration.

Nathanael Arkor (Sep 17 2020 at 15:02):

Cockett and Lemay have a paper called Cartesian Integral Categories and Contextual Integral Categories.

Nathanael Arkor (Sep 17 2020 at 15:02):

(There's a talk here, too.)

Paolo Perrone (Sep 17 2020 at 15:56):

The derivative can be seen as a functor, in that sense it maps functions to functions.
I've explained this a little bit in my notes here, Example 1.3.25. (I'm sure there are more specific references on this, though.)
I don't know about integration of functions. Anyone? (There are measure monads, but that's a bit different.)

JS PL (he/him) (Sep 18 2020 at 10:51):

James Wood said:

Did these authors ever cover integration? I feel as if I have a decent abstract understanding of differentiation, but that understanding is perhaps in conflict with integration.

I've worked on the integral side of the story. Thanks @Nathanael Arkor for sharing my paper with Robin. You might also be interested in these papers:
https://arxiv.org/abs/1707.08211, the first paper on integral categories
https://arxiv.org/pdf/1902.04555.pdf, where we give an integral category structure from integrating smooth functions
I also have some notes on "Cartesian differential categories with antiderivatives", if you're interested.

JS PL (he/him) (Sep 18 2020 at 10:52):

But the integral story is far from being finished! Still lots to do.

JS PL (he/him) (Sep 18 2020 at 10:55):

Scott Lee said:

What's the category theoretic understanding of differentiation and integration in calculus?

As @Nathanael Arkor pointed out, you might be interested in differential categories. Last summer I gave a talk introducing the first levels of differential categories:
https://pages.cpsc.ucalgary.ca/%7Erobin/FMCS/FMCS2019/slides/Simon-LemaySlidesFMCS2019.pdf
Always happy to talk differential categories! There are others on this board who work on differential categories like @Ben MacAdam, @Jonathan Gallagher , @Geoff Cruttwell , etc.